1 THE METHOD OF SUCCESSIVE APPROXIMATIONS 265
existence theorem to functions of several variables; it is usually formulated
in a more geometric way, as an existence theorem of manifolds having at each
point a given "tangent space." We will deal with that formulation in
Chapter XVIII.
It goes without saying that as usual we have expressed all results for vector
valued functions, so that, for instance, we practically never speak of" systems "
of equations; it is one of the virtues of the "vector space methods " that one
never need consider more than one equation, at least for the proofs of the
general theorems.
1. THE METHOD OF SUCCESSIVE APPROXIMATIONS
As in Chapter IX, K will denote either the real or the complex field, and
whenever a statement is made about Banach spaces without specifying K, it
is understood that all the Banach spaces concerned are over the same field.
(10.1.1) Let E, F be two Banach spaces, U (resp. V) an open ball in E (resp.F)
of center 0 and radius a (resp. jS). Let v be a continuous mapping ofUxV
into F, such that \\v(x, *) - v(x, y2)\\ < k • || yl - y2\\for xeU,yleV,y2e V,
where k is a constant such that Q^k<l. Then, if \\v(x, 0)|| < p(l - k) for
any x e U, there exists a unique mapping f of U into V such that
(10.1.1.1) /(*) = !>(*,/(*))
for any x e U; and f is continuous in U.
For any x e U, we will show there exists a sequence (yn) of points of V
such that jo = 0, yn = v(x, yn-i) for any n^l. We have to show that
if yp is defined and in V for 1 < p ^ n, then v(x, yn) e V. But we
have then for 2^p^n, yp — yp^l = v(x, yf^) - v(x9 yp-2\ hence
\\yp — yp-i\\ < k • ||j>p_i — yp-2\\> an(* by induction on p we conclude that
Ibp-JWll^'Mbill- Hence
(10.1.1.2) \\yp\\ < (1 + k + k2 + - • • -1- k>~l)\\yi\\ ^ 11*11/0 -k)<fi
which proves our contention. Moreover, by induction on n, we can write
yn —fn(x}^ where/, is a continuous mapping of U into V (3.11.5). We have
furthermore \\fn(x)-fn^(x)\\ ^kn'^(l - k) for *eU, hence the series
(/» — /«-1) is normally convergent (Section 7.1) in ^F(U); as F is complete, the